Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- Part I Near-equilibrium critical dynamics
- 1 Equilibrium critical phenomena
- 2 Stochastic dynamics
- 3 Dynamic scaling
- 4 Dynamic perturbation theory
- 5 Dynamic renormalization group
- 6 Hydrodynamic modes and reversible mode couplings
- 7 Phase transitions in quantum systems
- Part II Scale invariance in non-equilibrium systems
- Index
- References
6 - Hydrodynamic modes and reversible mode couplings
from Part I - Near-equilibrium critical dynamics
Published online by Cambridge University Press: 05 June 2014
Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- Part I Near-equilibrium critical dynamics
- 1 Equilibrium critical phenomena
- 2 Stochastic dynamics
- 3 Dynamic scaling
- 4 Dynamic perturbation theory
- 5 Dynamic renormalization group
- 6 Hydrodynamic modes and reversible mode couplings
- 7 Phase transitions in quantum systems
- Part II Scale invariance in non-equilibrium systems
- Index
- References
Summary
Equipped with the field theory representation of non-linear Langevin equations, the tools of dynamic perturbation theory, and the dynamic renormalization group introduced in Chapters 4 and 5, we are now in the position to revisit models for dynamic critical behavior that entail reversible mode couplings and other conserved hydrodynamic modes. We have already encountered some of these in Section 3.3. In models C and D, respectively, a non-conserved or conserved n-component order parameter is coupled to a conserved scalar field, the energy density. Through a systematic renormalization group analysis, we may critically assess the earlier predictions from scaling theory, and discuss the stability of fixed points characterized by strong dynamic scaling, wherein the order parameter and conserved non-critical mode fluctuate with equal rates, and weak dynamic scaling regimes, where these characteristic time scales differ. Next we investigate isotropic ferromagnets (model J), with the conserved spin density subject to reversible precession in addition to diffusive relaxation. Exploiting rotational invariance, we can now firmly establish the scaling relation z = (d + 2 − η)/2. Similar symmetry arguments yield a scaling relation for the dynamic exponents associated with the order parameter and the non-critical fields in the O(n)-symmetric SSS model that encompasses model E for planar ferromagnets and superfluid helium 4 (for n = 2), and model G for isotropic antiferromagnets (n = 3). There exist competing strong- and weak-scaling fixed points, with the former stable to one-loop order, and characterized by z = d/2 for all slow modes.
- Type
- Chapter
- Information
- Critical DynamicsA Field Theory Approach to Equilibrium and Non-Equilibrium Scaling Behavior, pp. 207 - 250Publisher: Cambridge University PressPrint publication year: 2014
References
, , and , 1976, Renormalized field theory of critical dynamics, Z. Phys. B Cond. Matt. 24, 113–127.Google Scholar
, , , and 1991, Dynamics of longitudinal and transverse fluctuations above Tc in EuS, Phys. Rev. B 43, 8755–8758.CrossRefGoogle ScholarPubMed
and , 1975, Field-theoretic techniques and critical dynamics. II. Ginzburg—Landau stochastic models with energy conservation, Phys. Rev. B 12, 4954—1962.CrossRefGoogle Scholar
, , , , and , 1998, Critical behavior of the three-dimensional Heisenberg antiferromagnet RbMnF3, Phys. Rev. B 57, 5281–5290.CrossRefGoogle Scholar
and , 1977, Deviations from dynamic scaling in helium and antiferromagnets, Phys. Rev. Lett. 38, 505–508.CrossRefGoogle Scholar
and , 1978, Field-theory renormalization and critical dynamics above Tc: helium, antiferromagnets, and liquid-gas systems, Phys. Rev. B 18, 353–376.CrossRefGoogle Scholar
, 2006, Model F in two-loop order and the thermal conductivity near the superfluid transition of 4He, Phys. Rev. B 73, 092503-1–4.CrossRefGoogle Scholar
and , 1980, Nonasymptotic critical dynamics near the superfluid transition in 4He, Z. Phys. B Cond. Matt. 40, 79–93.CrossRefGoogle Scholar
and , 2003, Critical dynamics of model C resolved, Phys. Rev. Lett. 91, 030601-1-4.CrossRefGoogle ScholarPubMed
and , 2004, Critical dynamics of stochastic models with energy conservation (model C), Phys. Rev. E 69, 036101-1–18.CrossRefGoogle Scholar
and , 2006, Critical dynamics: a field theoretical approach, J. Phys. A: Math. Gen. 39, R207–R313.CrossRefGoogle Scholar
and , 1976, Critical dynamics of isotropic antiferromagnets using renormalization-group methods: T ≥ TN, Phys. Rev. B 13, 4967–4983.Google Scholar
and , 2012, Critical dynamics of an isothermal compressible nonideal fluid, Phys. Rev. E 86, 061119-1–15.CrossRefGoogle ScholarPubMed
and , 1975, Critical transport anomalies in 4 – ∈ and 6 – ∈ dimensions, J. Phys. A: Math. Gen. 8, L9–L12.CrossRefGoogle Scholar
and , 1976, Renormalization group equations in critical dynamics. II. Binary liquids, superfluid helium and magnets, Progr. Theor. Phys. 56, 61–76.CrossRefGoogle Scholar
, , and , 1974, Renormalization-group methods for critical dynamics: I. Recursion relations and effects of energy conservation, Phys. Rev. B 10, 139–153.CrossRefGoogle Scholar
, , and , 1976, Renormalization-group methods for critical dynamics: II. Detailed analysis of the relaxational models, Phys. Rev. B 13, 4119–4131.CrossRefGoogle Scholar
, , and , 1974, Renormalization-group calculations of divergent transport coefficients at critical points, Phys. Rev. Lett. 32, 1289–1292.CrossRefGoogle Scholar
, , and , 1976, Renormalization-group treatment of the critical dynamics of superfluid helium, the isotropic antiferromagnet, and the easy-plane ferromagnet, Phys. Rev. B 13, 1299–1328; err. Phys. Rev. B 21, 2044-2045 (1980).CrossRefGoogle Scholar
and , 1977, Theory of dynamic critical phenomena, Rev. Mod. Phys. 49, 435–479.CrossRefGoogle Scholar
, 1977, Renormalized field theory for the critical dynamics of O(n)-symmetric systems, Z. Phys. B Cond. Matt. 26, 187–189.Google Scholar
, 1979, Field-theoretic methods applied to critical dynamics, in: Dynamical Critical Phenomena and Related Topics, ed. , Lecture Notes in Physics, Vol. 104, Heidelberg: Springer-Verlag, 26-–47.Google Scholar
, 1988, Universality of the dipolar dynamic crossover of cubic ferromagnets above Tc, Phys. Rev. B 38, 12027–12030.CrossRefGoogle ScholarPubMed
and , 1974, Critical dynamics of ferromagnets in 6 — ∈ dimensions, Phys. Rev. Lett. 33, 1383–1385.CrossRefGoogle Scholar
and , 1975, Critical dynamics of ferromagnets in 6 — ∈ dimensions: general discussion and detailed calculation, Phys. Rev. B 11, 4077—1100.CrossRefGoogle Scholar
and , 1977, Dynamic critical properties of a stochastic n-vector model, Physica 87A, 1–34.Google Scholar
, , , , and , 2000, Critical fluctuations in the weak itinerant ferromagnet Ni3Al: a comparison between self-consistent renormalization and mode-mode coupling theory, Phys. Rev. B 62, 1083–1088.CrossRefGoogle Scholar
, , and , 1976, Renormalization-group treatment of the critical dynamics of the binary-fluid and gas-liquid transitions, Phys. Rev. B 13, 2110–2123.CrossRefGoogle Scholar
and , 2003, Critical dynamics of the simple-cubic Heisenberg antiferromagnet RbMnF3: extrapolation to q = 0, Phys. Rev. B 67, 104411-1–6.CrossRefGoogle Scholar
, 1993, The Field Theoretic Renormalization Group in Critical Behavior Theory and Stochastic Dynamics, Boca Raton: Chapman & Hall / CRC, chapter 5.Google Scholar
, 1993, Quantum Field Theory and Critical Phenomena, Oxford: Clarendon Press, chapter 34.Google Scholar
, 1996, Critical dynamics of systems with reversible mode coupling terms: spherical limit, Europhys. Lett. 34, 525–530.CrossRefGoogle Scholar
, , and , 2010, Sound attenuation near the demixing point of binary liquids: interplay of critical dynamics and noncritical kinetics,Rep. Prog. Phys. 73, 066601, 1–36.CrossRefGoogle Scholar
and , 1997, Self-consistent screening approximation for critical dynamics, J. Phys. A: Math. Gen. 30, 3333–3343.CrossRefGoogle Scholar
and , 1995, Principles of Condensed Matter Physics, Cambridge: Cambridge University Press, chapter 8.CrossRefGoogle Scholar
, , , and , 2005, Universal dynamic exponent at the liquid–gas transition from molecular dynamics, Phys. Rev. Lett. 95, 255701-1–4.CrossRefGoogle ScholarPubMed
, , , , and , 2006, Critical dynamics in a binary fluid: simulations and finite-size scaling, Phys. Rev. Lett. 97, 025702-1–4.CrossRefGoogle Scholar
, 1987, Renormalization-group theory of critical phenomena near the lambda transition of 4He, J. Low Temp. Phys. 69, 51–75.CrossRefGoogle Scholar
, 1991, Renormalization-group flow equations of model F, Phys. Rev. B 44, 2697–2712.CrossRefGoogle ScholarPubMed
and , 1982, Nonlinear dynamic renormalization-group analysis above and below the lambda transition in 4He, Physica 109 & 110B, 1549–1556.
, , , and , 2013, Dynamic universality class of Model C from the functional renormalization group, Phys. Rev. B 88, 174301-1–4.CrossRefGoogle Scholar
and , 2011, Transport phenomena in fluids: finite-size scaling for critical behavior, Europhys. Lett. (EPL) 94, 36001-1–6.